Idea

A topological space may have very little separating it from ‘manifoldness’, yet a ‘singularity’ can cause havoc! The simple example, here, is known as the Warsaw Circle as it was studied extensively by K. Borsuk and his Polish collaborators, see the book ([Borsuk]).

Definition

The Warsaw circleSWS_W is the subset of the plane, ℝ2\mathbb{R}^2, specified by

Note There is a variant version SW′S_W' with no (x,sin(1x))(x,\sin(\frac{1}{x}))-bit for the x<0x\lt 0 and the curve CC joins (0,0)(0,0) to (12π,0)(\frac{1}{2\pi}, 0). The discussion adapts very easily to that. For this version, there is a surjective continuous map ℝ→SW′\mathbb{R} \to S_W'. See eg Wikipedia for a picture.

There is a simple continuous map from S0S^0, the 0-circle, {−1,1}\{-1,1\}, to SWS_W which is a weak homotopy equivalence. (For instance define f(−1)=(0,0)f(-1) = (0,0) and f(1)f(1) to be any point in the outer sin(1/x)sin(1/x) part of the space, it does not matter which one.) This is not a homotopy equivalence. (In fact it is instructive to look at maps from SWS_W to S0S^0! It does not take long.)

A striking thing about the picture is that it ‘clearly’ divides the plane into two components, an inside and an outside, and has a definite sense of being ‘almost’ a circle. It has a line of singularities, but otherwise … .

If we consider, not just SWS_W as a compact metric space, but as a subspace of the plane, then we can take small open neighbourhoods of SWS_W, to be definite take

This looks like an annulus with a thickenning at one small section. It has the homotopy type of a circle. If N>nN \gt n, N1N(SW)⊂N1n(SW)N_{\frac{1}{N}}(S_W)\subset N_{\frac{1}{n}}(S_W), of course, (we will write inNi^N_n for this map, and this is a homotopy equivalence. The Warsaw circle, SWS_W, is clearly the intersection of all these almost annular neighbourhoods. (Note, also clearly, that the complements of these neighbourhoods are gradually occupying more and more of the two components of ℝ2−SW\mathbb{R}^2- S_W.)

We have a inverse system (pro-object) of topological spaces all of which have the homotopy type of a polyhedron, … in fact always the same polyhedron, S1S^1. Note that by our use of a specific cofinal family of neighbourhoods of SWS_W, indexed by the natural numbers, we have an inverse sequence. That was a choice and we could have chosen differently or not at all. The ability to pick a sequence of neighbourhoods is related to the fact that we are considering a compact metric space.

Another point to note is that not only is each of the neighbourhoods homotopic to S1S^1, but the inclusion maps making up the ‘bonds’ of the inverse sequence, are homotopy equivalence. This is a particularity of SWS_W and other examples, such as the solenoids need not have this ‘stability’ property. The Warsaw circle is an example of what is called a stable space.

A (Borsuk) shape map f:S1→SWf\colon S^1 \to S_W

There is a sequence of maps, {fn:S1→N1n(SW)∣n∈ℕ}\{f_n : S^1 \to N_{\frac{1}{n}}(S_W)\mid n\in \mathbb{N}\}, so that for each pair, (n,N)(n,N), with N>nN\gt n, there is a homotopy fn∼inNfNf_n \sim i^N_n f_N. This makes a (Borsuk) shape map from the circle to the Warsaw circle. Each fnf_n is in fact a homotopy equivalence and we can use a choice of homotopy inverses to get another shape map g:SW→S1g : S_W\to S^1 and these make up a shape equivalence.

(A more detailed description of shape maps and shape equivalences in the Borsuk version of shape theory, is given in the entry Borsuk shape theory. The version given here skates over some points. It is, in fact, near the ANR-systems approach to shape.)

From a Čech point of view

To get Cˇ(SW,−)\check{C}(S_W,-) Čech nerve complex of SWS_W, (see Čech methods), we can calculate Cˇ(SW,α)\check{C}(S_W,\alpha) for an arbitrary open coverα\alpha of SWS_W, but we need not do that (in fact that is a silly thing to do!). We first note that SWS_W is compact so we need only consider finite open covers, as these form a cofinal subcategory of the category of all open covers. (‘Cofinal subcategory’ means that its inclusion into the bigger category is a cofinal functor.)

Next we look at any finite open cover and note that it has a refinement in the form of open balls of radius 1n\frac{1}{n}, in other words we can restrict to (well chosen) such covers, giving a countable family of open covers that have to be worked with.

There may be fine detail in the rectangle depending on the choice of cover, but that detail will disappear as one passes to finer and finer scales. (New holes may occur, but again going finer those disappear.) Cofinally it looks like a space obtained by adding in a thin rectangle transverse to a circle at one small segment. For different open coverings, the only difference will be where the region of attachment (marked **) will occur and the relative thinness of the rectangle. The line of singularities given by the interval [−1,1][-1,1] on the yy-axis cannot be ‘observed’, of course. If one passes to finer and finer covers, most of the curve does not change appreciably. It just gets subdivided, but the part near ** will lengthen, ‘spawning’ a very large number of new vertices.

There are two important points to note:

(essentially) each Cˇ(SW,α)\check{C}(S_W,\alpha) has the homotopy type of a circle

the transition maps, C(SW,α)→C(SW,β)C(S_W,\alpha)\to C(S_W,\beta), will be (cofinally) homotopy equivalences.

(With a bit more care in the choice of the covers these can be made exact statements, not just ‘essentially’ or cofinally true.)

We note that there are obvious maps of pro-objects Cˇ(S1,−)→Cˇ(SW,−)\check{C}(S^1,-)\to \check{C}(S_W,-), and back again. These give an isomorphism in pro−Ho(sSets)pro-Ho(sSets). This is the Čech homotopy versions of the observations made for Borsuk’s shape above.